Describe the horizontal and vertical asymptotes of f(x)=log4x

Question

.(1 point)
Responses

The function f(x)
has a horizontal asymptote at y=4
and a vertical asymptote at x=0
.
The function f left parenthesis x right parenthesis has a horizontal asymptote at y equals 4 and a vertical asymptote at x equals 0 .

The function f(x)
has a horizontal asymptote at y=0
and a vertical asymptote at x=4
.
The function f left parenthesis x right parenthesis has a horizontal asymptote at y equals 0 and a vertical asymptote at x equals 4 .

The function f(x)
has a vertical asymptote at x=0
and no horizontal asymptote.
The function f left parenthesis x right parenthesis has a vertical asymptote at x equals 0 and no horizontal asymptote.

The function f(x)
has a horizontal asymptote at y=0
and no vertical asymptote.

Answer 1

To analyze the function \( f(x) = \log_4(x) \):

1. Vertical Asymptote: A logarithmic function has a vertical asymptote where the argument of the log function approaches zero from the right. Therefore, for \( f(x) = \log_4(x) \), there is a vertical asymptote at \( x = 0 \) since the logarithm is undefined for \( x \leq 0 \).

2. Horizontal Asymptote: A logarithmic function grows without bound as \( x \) increases. Specifically, as \( x \to \infty \), \( f(x) = \log_4(x) \to \infty \) as well. Therefore, there is no horizontal asymptote because the function does not approach a constant value as \( x \) increases.

Given this analysis, the correct response is:

The function f(x) has a vertical asymptote at x=0 and no horizontal asymptote.